Bisubmodular Functions Research Articles

The Box Convolution and the Dilworth Truncation of Bisubmodular Functions

The Box Convolution and the Dilworth Truncation of Bisubmodular Functions

A Bisubmodular Approach to Event Detection and Prediction in Multivariate Social Graphs

A burst event on a social graph is usually framed as an anomalous and unexpected pattern that is characterized as a compact or correlated subset of affected vertices, which is a subgraph. Subgraph detection becomes a serious problem when social graphs involve multiple attributes (i.e., multivariate graph). Most existing methods are not capable of handling the feature selection and subgraph detection problems simultaneously on the multivariate graph. In this article, we propose multivariate anomalous subgraph scanning (MASS), a generic model that detects anomalous events on the multivariate social graph. First, we reformulate the traditional nonparametric statistics as a new statistical objective function that simultaneously measures the significance of a vertices subset and an attributes subset to generate an indicator of ongoing or upcoming events. Then, we reformulate the objective function as the difference between two bisubmodular functions and approximate it with a bisubmodular objective function, which can be optimized in linear time, with an analysis of its theoretical properties. We demonstrate the performance of our proposed method using two burst event detection and prediction tasks from the real world.

A compact representation for minimizers of k-submodular functions

A k-submodular function is a generalization of submodular and bisubmodular functions. This paper establishes a compact representation for minimizers of a k-submodular function by a poset with inconsistent pairs (PIP). This is a generalization of Ando–Fujishige’s signed poset representation for minimizers of a bisubmodular function. We completely characterize the class of PIPs (elementary PIPs) arising from k-submodular functions. We give algorithms to construct the elementary PIP of minimizers of a k-submodular function f for three cases: (i) a minimizing oracle of f is available, (ii) f is network-representable, and (iii) f arises from a Potts energy function. Furthermore, we provide an efficient enumeration algorithm for all maximal minimizers of a Potts k-submodular function. Our results are applicable to obtain all maximal persistent labelings in actual computer vision problems. We present experimental results for real vision instances.

Parametric bisubmodular function minimization and its associated signed ring family

The present paper shows an extension of the theory of principal partitions for submodular functions to that for bisubmodular functions. We examine the structure of the collection of all solutions of a parametric minimization problem described by a bisubmodular function and two vectors. The bisubmodular function to be minimized for each parameter is the sum of the bisubmodular function and a parameterized box-bisubmodular function given in terms of the two vectors. We show that the collection of all the minimizers for all parameters forms a signed ring family and it thus induces a signed poset on a signed partition of the underlying set. We further examine the structure of the signed ring family and reveal the decomposition structure depending on critical values of the parameter. Moreover, we discuss the relation between the results of this paper on bisubmodular functions and the theory of principal partitions developed for submodular functions.

Maximizing k -Submodular Functions and Beyond

We consider the maximization problem in the value oracle model of functions defined on k -tuples of sets that are submodular in every orthant and r -wise monotone, where k ⩾ 2 and 1 ⩽ r ⩽ k . We give an analysis of a deterministic greedy algorithm that shows that any such function can be approximated to a factor of 1/(1 + r ). For r = k , we give an analysis of a randomized greedy algorithm that shows that any such function can be approximated to a factor of 1/(1+√ k /2. In the case of k = r = 2, the considered functions correspond precisely to bisubmodular functions, in which case we obtain an approximation guarantee of 1/2. We show that, as in the case of submodular functions, this result is the best possible both in the value query model and under the assumption that NP ≠ RP . Extending a result of Ando et al., we show that for any k ⩾ 3, submodularity in every orthant and pairwise monotonicity (i.e., r = 2) precisely characterize k -submodular functions. Consequently, we obtain an approximation guarantee of 1/3 (and thus independent of k ) for the maximization problem of k -submodular functions.

A Min-Max Theorem for Transversal Submodular Functions and Its Implications

Huber and Kolmogorov [Towards minimizing $k$-submodular functions, in Proceedings of ISCO 2012, Lecture Notes in Comput. Sci. 7422, Springer, Heidelberg, 2012, pp. 451--462] introduced a concept of $k$-submodular function as a generalization of ordinary submodular (set) functions and bisubmodular functions and obtained a min-max theorem for the minimization of $k$-submodular functions. Also Kuivinen [Discrete Optim., 8 (2011), pp. 459--477] considered submodular functions on (product lattices of) diamonds and showed a min-max theorem for the minimization of submodular functions on diamonds. In the present paper we consider a common generalization of $k$-submodular functions and submodular functions on diamonds, which we call a transversal submodular function (a t-submodular function, for short). We show a min-max theorem for the minimization of t-submodular functions in terms of a new norm composed of $\ell_1$ and $\ell_\infty$ norms. This reveals a relationship between the obtained min-max theorem and that for the minimization of ordinary submodular set functions due to Edmonds [Submodular functions, matroids, and certain polyhedra, in Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications, R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds., Gordon and Breach, New York, 1970, pp. 69--87].We also show how our min-max theorem for t-submodular functions can be used to prove the min-max theorem for $k$-submodular functions by Huber and Kolmogorov and that for submodular functions on diamonds by Kuivinen. Moreover, we show a counterexample to a characterization, given by Huber and Kolmogorov [Towards minimizing $k$-submodular functions, in Proceedings of ISCO 2012, Lecture Notes in Comput. Sci. 7422, Springer, Heidelberg, 2012, pp. 451--462], of extreme points of the $k$-submodular polyhedron and make it a correct one by fixing a flaw therein.

Oracle Tractability of Skew Bisubmodular Functions

In this paper we consider skew bisubmodular functions as recently introduced by the authors and Powell. We construct a convex extension of a skew bisubmodular function which we call Lovász extension in correspondence to the submodular case. We use this extension to show that skew bisubmodular functions given by an oracle can be minimized in polynomial time.

Generalized skew bisubmodularity: A characterization and a min–max theorem

Huber, Krokhin, and Powell (2013) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain. In this paper we consider a natural generalization of the concept of skew bisubmodularity and show a connection between the generalized skew bisubmodularity and a convex extension over rectangles. We also analyze the dual polyhedra, called skew bisubmodular polyhedra, associated with generalized skew bisubmodular functions and derive a min–max theorem that characterizes the minimum value of a generalized skew bisubmodular function in terms of a minimum-norm point in the associated skew bisubmodular polyhedron.

Generalized roof duality and bisubmodular functions

Consider a convex relaxation fˆ of a pseudo-Boolean function f. We say that the relaxation is totally half-integral if fˆ(x) is a polyhedral function with half-integral extreme points x, and this property is preserved after adding an arbitrary combination of constraints of the form xi=xj, xi=1−xj, and xi=γ where γ∈{0,1,12} is a constant. A well-known example is the roof duality relaxation for quadratic pseudo-Boolean functions f. We argue that total half-integrality is a natural requirement for generalizations of roof duality to arbitrary pseudo-Boolean functions.Our contributions are as follows. First, we provide a complete characterization of totally half-integral relaxations fˆ by establishing a one-to-one correspondence with bisubmodular functions. Second, we give a new characterization of bisubmodular functions. Finally, we show some relationships between general totally half-integral relaxations and relaxations based on the roof duality.On the conceptual level, our results show that bisubmodular functions provide a natural generalization of the roof duality approach to higher-order terms. This can be viewed as a non-submodular analogue of the fact that submodular functions generalize the s-t minimum cut problem with non-negative weights to higher-order terms.

Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization

Bisubmodular functions are a natural "directed", or "signed", extension of submodular functions with several applications. Recently Fujishige and Iwata showed how to extend the Iwata, Fleischer, and Fujishige (IFF) algorithm for submodular function minimization (SFM) to bisubmodular function minimization (BSFM). However, they were able to extend only the weakly polynomial version of IFF to BSFM. Here we investigate the difficulty that prevented them from also extending the strongly polynomial version of IFF to BSFM, and we show a way around the difficulty. This new method gives the first combinatorial strongly polynomial algorithm for BSFM. This further leads to extending Iwata's fully combinatorial version of IFF to BSFM.

Bipolarization of posets and natural interpolation

The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of [ 0 , 1 ] n . We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be seen as particular instances of this scheme. Lastly, an application to multicriteria aggregation with multiple reference levels illustrates all the results presented in the paper.

Bisubmodular Function Minimization

This paper presents the first combinatorial polynomial algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. Since the rank functions of delta-matroids are bisubmodular, the scaling algorithm naturally leads to the first combinatorial polynomial algorithm for testing membership in delta-matroid polyhedra.

A characterization of bisubmodular functions

Given a nonempty finite set E, let 3 E be the set of all the ordered pairs of disjoint subsets of E. We call a function on 3 E a biset function. A biset function f:3 E → R is called bisubmodular if we have ∀(X 1,Y 1), (X 2,Y 2) ϵ 3 E: ƒ(X 1,Y 1) + ƒ(X 2,Y 2)⩾ƒ(X 1 ∪ Y 2) − (Y 1 ∪ Y 2),(Y 1 ∪ Y 2) − (X 1 ∪ x 2)) +ƒ(X 1 ∩ X 2, Y 1 ∩ Y 2). We give a simple necessary and sufficient condition for a biset function to be bisubmodular.

Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra

This paper relates an axiomatic generalization of matroids, called a jump system, to polyhedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron but form a subset of them. However, it is shown that the convex hull of the set of points of a jump system is a bisubmodular polyhedron, and that the integral points of an integral bisubmodular polyhedron determine a (special) jump system. The authors prove addition and composition theorems for jump systems, which have several applications for delta-matroids and matroids.

The Orthant Non-Interaction Theorem for Certain Combinatorial Polyhedra and its Implications in the Intersection and the Dilworth Truncation of Bisubmodular Functions

We show a theorem characterizing a class of polyhedra that are expressed by systems of inequalities with {0,±1} coefficients and have no interaction between maximal faces in ditrerent orthants w.r.t. the defin-ing inequalities. Based on this theorem we give a sufficient condition for the intersection of two bisub-modular polyhedra to be integral. We also extend the notion of the Dilworth truncation of bisubmodular functions to wider plausible class of bisubmodular functions. i.e.. newly defined “compliant-intersecting bisubmodular” functions

Bi-submodular functions, distributive valuations, and distance functions in lattices

Bi-submodular functions, distributive valuations, and distance functions in lattices